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Refractive index dispersion and Drude model Optics, Eugene Hecht, Chpt. 3
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Dielectrics Electric field is reduced inside dielectric –Space charge partly cancels –E / E v = / 0 Also possible for magnetic fields –but usually B = B v and = 0 Result: light speed reduced v = c ( ) = c/n < c Wavelength also reduced = 0 /n E-field Dielectric Index of refraction: n
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Conventions Polarization of materials Separate into material and vacuum parts – E = 0 E + P –linear material: P = 0 E Material part is due to small charge displacement Similar equation for magnetic polarization – B / = B / 0 + M Most optical materials have = 0 Refractive index n 2 = ( / 0 ) ( / 0 ) = [1 + P / ( 0 E)] / [1 + 0 M/B] Drop magnetic part n 2 = [1 + P / ( 0 E)]
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Material part of polarization Polarization due to small displacements Examples: –Polar molecules align in field –Non-polar molecules – electron cloud distorts Optical frequencies –Nucleus cannot follow fast enough Too heavy –Consider mainly electron cloud Distorted electron cloud
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Model of atom Lowest order – everything is harmonic oscillator Model atom as nucleus and electron connected by spring Newton’s law: F = m a Spring restoring force: F R = - k x = - m 2 x –Resonant freq of mass-spring: = k/m Driving force: F D = q e E Damping force: F = - m v Resultant equation: q e E - m dx/dt - m 2 x = m d 2 x/dt 2 Free oscillation: (E=0, =0) –d 2 x/dt 2 + 2 x = 0 Use complex representation for E –E = E 0 e i t Forced oscillation: –motion matched drive frequency –x = x 0 e i t Result: x 0 = (q/m) E 0 / [ 0 2 - 2 + i ]
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Refractive index & dispersion Drude model Polarization of atom –Define as charge times separation –P A = q e x Material has many atoms: N Material polarization: P = q e x N Recall previous results n 2 = [1 + P / ( 0 E)] x 0 = (q/m) E 0 / [ 0 2 - 2 + i ] Result is dispersion equation: Correction for real world complications: Sum over all resonances in material f is oscillator strength of each transition ~ 1 for allowed transition
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Sample materials Polar materials Refractive index approx. follows formula Resonances in UV Polar materials also have IR resonances –Nuclear motion – orientation
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Anomalous dispersion Above all resonance frequencies Dispersion negative Refractive index < 1 v > c X-ray region
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Metals and plasma frequency “Free” conduction electrons – resonance at zero 0 = 0 Metals become transparent at very high frequency – X-ray Neglect damping At low frequency n 2 < 0 –refractive index complex –absorption At high frequency –n becomes real –like dielectric –transparency Plasma freq
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Skin depth in metals Electrons not bound Current can flow Conductance ~ 1/R causes loss Maxwell’s equations modified Wave solution also modified –Express as complex refractive index –n complex = n R – i c / (2 ) –E = E 0 e - z/2 e i(kz- t) Result for propagation in metal: I = I 0 e - z, 1/ = skin depth Metals: 1/ << Example copper: – = 100 nm, 1/ = 0.6 nm = / 170 – = 10 m, 1/ = 6 nm = / 1700 – = 10 mm, 1/ = 0.2 m = / 50,000 – 1/ ~ Similar to n >> 1 Strong reflection – not much absorption Metal Density Ro f skin depth (microOhm cm) (GHz) (microns) Aluminum 2.70 g/cc2.824;478.59 0.12 Copper 8.89 g/cc1.7241;409.1 0.1033 Gold 19.3 g/cc2.44;403.8 0.12 Mercury 13.546 g/cc 95.783;10,975.0.15 Silver 10.5 g/cc1.59;2600.12 Drude -- low frequency limit 0
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Reflectivity of metals Assume perfect conductor No electric field parallel to interface Reflectivity at normal incidence (assume n i = 1) Power reflected R = r r* 1 for large absorption E field incident reflected metal Standing wave -- zero at surface Normal incidence reflection from metal
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Plasmons Assume 0 = 0 for conduction electrons -- keep damping Transition occurs when optical frequency exceeds collision frequency –depends on dc resistivity –lower resistivity = higher frequency transition Above collision frequency -- Plasmons Plasmons quenched at plasma frequency Example -- silver – = 6.17 x 10 7 / -m, plasma = 9.65 x 10 14 Hz (311 nm, 4 eV) – e = 1/(13 fs) = 7.7 x 10 13 Hz –plasmons beyond ~ 23.5 microns wavelength
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Plasmons and nano optics Small metal particles can act like inductors, capacitors Maxwell’s equation for current density: –Separate into vacuum and metal parts Vacuum (or dielectric) part is capacitor Metal part is inductor plus series resistor RLC circuit parameters –Resonance frequency 0 =1/sqrt(LC) = plasma –Resonance width = R/L = collision Structure geometry can increase L and C –Strong local field enhancement possible in capacitor conductivity Displacement current Vacuummetal metal dielectric LC Nano optic RLC circuit
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“Left hand” materials: (E in plane of incidence) Sign of and both negative Strange properties Refraction backward Example -- E parallel, P-polarization Two components of E Parallel to surface –E i cos i + - E r cos r = E t cos t Perpendicular to surface –1. Space charge attenuates E t – i E i sin i + r E r sin r = t E t sin t –Sign of t is negative –2. Use Snell’s law –n i E i + n r E r = n t E t B is parallel to surface –same as perpendicular E r parallel = (n t cos i - n i cos t ) / (n t cos i + n i cos t ) t parallel = (2n i cos i ) / (n t cos i + n i cos t ) Interface ii rr tt EiEi ErEr EtEt E’ t ’t’t nini ntnt Propagation direction E x B Momentum
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Left handed materials - fabrication Need sign of and both negative Problem: magnetic part usually ~1 Solution: Fool the EM field –LC circuit – material in capacitor gap indirectly modifies magnetic material Loops are inductors Gap is capacitor LC circuit E B k Artificial “left-hand” material
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